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How should the following statement be interpreted: Let $f$ be a function with range in* $[0,1]$. (Here the word "in" has been deliberately included.)

In this context, the range is the same as the image.

I've always taken the statement to mean that the image of $f$ lies in $[0,1]$. Are there alternate interpretations of this statement? For example, can the above statement also mean that the image of $f$ is exactly $[0,1]$?

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The most common interpretation of that phrase would be that the range (image) of $f$ is a subset of $[0,1]$. That is, we may say $f:D \to [0,1]$ if the domain of $f$ is $D$. That is, the codomain of $f$ may be taken to be $[0,1]$.

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